Method for determining the stator flux of an asynchronous machine

ABSTRACT

The invention relates to a method for determining the stator flux of an asynchronous machine when a stator current (i 2 ) and stator voltage (u s ) of the asynchronous machine are measured and a short-circuit inductance (σL s ), stator inductance (L s ) and rotor time constant (τ r ) of the machine are assumed to be known, wherein the product of the stator current (i s ) and a stator resistance estimate (R se ) is determined, the obtained product is subtracted from the stator voltage (u s ), and the obtained voltage difference is integrated with respect to time to obtain a first stator flux estimate (ψ se ). In the invention, the method further comprises the steps of determining a second stator flux estimate (ψ se ,id) on the basis of the stator voltage (u s ), the stator current (i s ), the short-circuit inductance (σL s ), the stator inductance (L s ), the rotor time constant (τ r ) and the stator resistance estimate (R se ); determining a stator flux correction term (Δψ se ) as a difference between the first stator flux estimate (ψ se ) and the second stator flux estimate (ψ se ,id), and summing the stator flux correction term (Δψ se ) weighted by a constant coefficient (w) with said voltage difference.

This invention relates to a method for determining the stator flux of anasynchronous machine when a stator current and stator voltage of theasynchronous machine are measured and a short-circuit inductance, statorinductance and rotor time constant are assumed to be known, wherein theproduct of the stator current and a stator resistance estimate isdetermined, the obtained product is subtracted from the stator voltage,and the obtained voltage difference is integrated with respect to timeto obtain a first stator flux estimate. The stator resistance estimateis also determined in the method.

Control of an asynchronous machine by the AC inverter technique oftenaims at a desired behaviour of the torque created by the machine whenthe current and voltage supplied to the machine are known. One attemptsto affect the electric torque, which is expressed as a function of thestator flux and stator current:

    T.sub.m =c(ψ.sub.s ×i.sub.s),                    (1)

where

T_(m) =electric torque,

c=constant coefficient,

ψ_(s) =stator flux, and

i_(s) =stator current.

Proper torque control thus requires that not only the current i_(s) butalso the stator flux of the machine or a parameter proportional to it(such as the rotor or air gap flux) is known. This is not a problem whenoperating at relatively high frequencies, in which case that stator fluxis obtained in a well-known manner by directly integrating the voltagesupplied to the machine: ##EQU1## where

u_(s) =stator voltage, and

ω_(s) =supply frequency.

According to Eq. 2, ψ_(s) is easy to calculate when the supply voltageand its frequency are known. It also appears from this equation thatwhen ω_(s) is decreased below a predetermined nominal frequency, thevoltage has to be decreased in order that ψ_(s) would not increaseexcessively and that the machine would not be saturated.

However, Eq. 2 is not usable at low frequencies as the voltage acting inthe windings of the machine actually deviates from the supply voltage bythe voltage drop occurring in the resistances of the windings. So theproportion of the drop component in the voltage increases as u_(s) hasto be decreased on decreasing ω_(s). Therefore the drop component shouldbe taken into account at low frequencies, that is, the flux should becalculated from the equation:

    ψ.sub.s =∫(u.sub.s -R.sub.s i.sub.s)dt,           (3)

where R_(s) =stator resistance.

However, the accuracy of the flux calculated by this equation dependsgreatly on the measuring accuracy and the accuracy of the applied R_(s)estimate. As a certain error is allowed in the flux, the accuracyrequirements of the stator resistance R_(s) and the voltage and currentmeasurements increase on approaching the zero frequency. At the zerofrequency, errors in the voltage to be integrated cause a cumulativeerror component to occur in the flux estimate calculated by Eq. 3. Inpractice, the RI compensation cannot therefore be used alone below 10 Hzwithout a considerable error in the flux estimate.

The problem can be avoided either by direct or indirect vector control.In the former case, the stator flux is measured directly by means of aprobe positioned in the machine, while in the latter method it iscalculated indirectly on the basis of the stator current and the speeddata obtained from a tachometer positioned on the machine shaft. In bothcases, the torque of the machine can be adjusted even at the zerofrequency, but the additional probe, which both methods require, isrelatively expensive and deteriorates the reliability.

For the time being, there does not exist any method based on merevoltage and current measurements by means of which the torque of anasynchronous machine could be adjusted accurately when the operatingfrequency is below 10 Hz. The problem is due to the fact that when usingconventional methods based on Eq. 3, such as that disclosed in U.S. Pat.No. 4,678,248, it is practically impossible to estimate the stator fluxat low values of ω_(s).

The above-mentioned problem can be avoided by using a method accordingto the present invention, which is characterized in that it comprises,in addition to the steps mentioned in the beginning, the steps of

determining a second stator flux estimate on the basis of the statorvoltage, the stator current, the short-circuit inductance, the statorinductance, the rotor time constant and the stator resistance estimate;

determining a stator flux correction term as a difference between thefirst stator flux estimate and the second stator flux estimate, and

summing the stator flux correction term weighted by a constantcoefficient with said voltage difference.

In the method of the invention, the stator flux estimate is calculatedby using Eq. 3 while correcting the voltage u_(s) -R_(s) i_(s) to beintegrated so as to compensate for the error created in the estimateduring integration. The direction of the corrections is determined bycalculating a second stator flux estimate on the basis of the changes ofthe values of the machine and subtracting the estimate obtained by Eq. 3from the second estimate, and small corrections are then made in thevoltage to be integrated in the direction of the obtained flux deviationso that the estimate calculated by Eq. 3 will be equal to the secondestimate on the average. Preferably, the second estimate is determinedby observing a differential equation defined by means of theabove-mentioned machine parameters so as to describe the operation ofthe machine in the neighbourhood of two different points in time forobtaining a pair of equations enabling the mathematical solution of thestator flux estimate, and by selecting the second stator flux estimateout of the two solutions of the pair of equations so that the selectedstator flux estimate is closer to the first stator flux estimate. Inconnection with the corrections, one also calculates the statorresistance estimate, which is needed in the calculations of the voltageu_(s) -R_(s) i_(s) and said second flux estimate. Preferably, thecalculation of the stator resistance estimate comprises the steps of

determining the scalar product of the stator flux correction term andthe stator current;

weighting the scalar product by multiplying it by a negative constantcoefficient; and

integrating the weighted scalar product with respect to time.

In the method according to the invention, the cumulative errors formedduring the integration of the voltage to be integrated are compensatedfor by the corrections made in the voltage, as a result of which arelatively accurate stator flux estimate will be obtained by means ofthe method according to the invention even on approaching the zerofrequency when the only measuring data are the stator voltage and statorcurrent.

In the following the method according to the invention will be describedin more detail with reference to the attached drawings, in which

FIGS. 1 and 2 show examples of a stator flux vector as a function oftime;

FIG. 3 is a block diagram illustrating by way of example a stator fluxcalculation method based on the observation of variation in the valuesof an asynchronous machine;

FIG. 4 is a block diagram illustrating a conventional stator fluxcalculation method based on Eq. 3; and

FIG. 5 is a block diagram illustrating by way of example a stator fluxcalculation method according to the invention for an asynchronousmachine.

The calculation of the above-mentioned second stator flux estimate isbased on the widely known differential and current equations of thestator and rotor of an asynchronous machine, which are in the coordinatesystem of the stator: ##EQU2## where

ψ_(r) =rotor flux,

i_(r) =rotor current,

ω_(m) =mechanical rotation rate,

R_(r) =rotor resistance,

L_(s) =stator inductance,

L_(r) =rotor inductance, and

L_(m) =primary inductance.

By using Eq. 6 and 7, the rotor flux and rotor current can be given bymeans of the stator flux and stator current: ##EQU3## where ##EQU4##

Introducing Eq. 8 and 9 into Eq. 5 gives ##EQU5## where τ_(r) =L_(r)/R_(r) =rotor time constant.

Introducing the expression of the stator flux derivative solved from Eq.4 into Eq. 10 gives ##EQU6##

This equation binds together the values of the stator (flux, current andvoltage) and the mechanical speed. The latter can be eliminated by firstmultiplying Eq. 11 by the complex conjugate of the vector ψ_(s) -σL_(s)i_(s) (cf. Eq. 8) parallel to the rotor flux and then taking the realparts from both sides of the obtained equation, which gives: ##EQU7##

As i_(s) and u_(s) are obtained by measuring, ψ_(s) is the only unknownvalue of Eq. 12, when it is assumed that the parameters σL_(s), τ_(r)and L_(s) +R_(s) τ_(r) are known. However, the problem is that ψ_(s) isa vector comprising both the real and the imaginary parts (or theamplitude and phase angle), and so there are two unknown parameters fora single equation. Accordingly, additional conditions are required forsolving ψ_(s).

One way of obtaining the above-mentioned additional conditions andsolving the flux would be to apply Eq. 12 at more than one point oftime. For example, one could select points of time t₀ and t₁, and so onecould utilize the current, current derivative and voltage measurementsperformed at the particular point of time. It would thereby be necessarythat Eq. 12 is true at both points of time, so that an equation pairwould be obtained for attempting to solve ψ_(s). However, this methodwould not be usable as such, as ψ_(s) (t₀) and ψ_(s) (t₁) would beunequal in a general case, so that the number of unknown parameterswould again be greater than that of the equations. In addition,practice, the derivative of the current cannot be measured accurately ata particular time, but variation in the current should be observed overa longer period of time.

In the present method, the above-mentioned problem is solved byobserving the average states of the machine over time intervals [t₀ -Δt,t₀ ] and [t₁ -Δt, t₁ ] having a duration Δt, where t₀ represents thepresent time and

    t.sub.0 -t.sub.1 =Δt.sub.1 >0                        (13)

Variation in the average values of the machine is examined on the basisof the history data having the duration Δt+Δt₁ on transition from thetime interval [t₁ -Δt, t₁ ] to the time interval [t₀ -Δt, t₀ ]. Thesituation is illustrated in FIG. 1, which shows an example of variationin the stator flux vector within the interval [t₁ -Δt, t₀ ]. In thefigures, Δt<t₁, but these time intervals may also be consecutive oroverlapping, so that Δt≧t₁.

The following objective is to deduce equations in which the timeaverages of the stator flux within said time intervals are given bymeans of ψ_(s) (t₀). When Eq. 12 is then applied separately to theaverage values of both intervals, a pair of equations is obtained, inwhich the present-time flux ψ(t₀) to be estimated is the only unknownparameter (except for the parameters of the machine).

Eq. 3 gives the following expression to the average stator flux actingwithin the time interval [t₀ -Δt, t₀ ]: ##EQU8## where ψ_(s),ave0 =thetime average of the stator flux within the interval [t₀ -Δt, t₀ ].

Eq. 14 is reduced by partial integration into: ##EQU9## where Δψ_(s0) isthe deviation of the present-time stator flux from its time averagewithin the interval [t₀ -Δt, t₀ ]: ##EQU10##

Correspondingly, the average stator flux within the time interval [t₁-Δt, t₁ ] will be: ##EQU11## where ψ_(s),ave1 =the time average of thestator flux within the interval [t₁ -Δt, t₁ ], and ##EQU12##

In these equations, Δψ_(s) represents the change of the stator flux ontransition from the point of time t₁ to the point of time t₀, andΔψ_(s1) represents the deviation of ψ_(s) (t₁) from the time average ofthe stator flux within the interval [t₁ -Δt, t₁ ] (FIG. 2).

Correspondingly, the time averages of the current derivative, thecurrent and the voltage within the respective intervals are: ##EQU13##where i'_(s),ave0 =the time average of the stator current derivativewithin the interval [t₀ -Δt, t₀ ],

i'_(s),ave1 =the time average of the stator current derivative withinthe interval [t₁ -Δt, t₁ ],

i_(s),ave0 =the time average of the stator current within the interval[t₀ -Δt, t₀ ],

i_(s),ave1 =the time average of the stator current within the interval[t₁ -Δt, t₁ ],

u_(s),ave0 =the time average of the stator voltage within the interval[t₀ -Δt, t₀ ], and

u_(s),ave1 =the time average of the stator voltage within the interval[t₁ -Δt, t₁ ].

In practice, the average values of the time interval [t₁ -Δt, t₁ ] arenot worth calculating separately (Eq. 19, 21, 23 and 25) as they can beobtained from the average values of the interval [t₀ -Δt, t₀ ] bydelaying. To prove this a delay operator D is now defined so that

    D(τ)f(t)=f(t-τ),                                   (26)

where f is an arbitrary function of the time t and τ (>0) represents anarbitrary delay by which f(t) is delayed when multiplied by D(τ).

Using this delay operator the time averages of the current derivativecan now be written within the intervals [t₀ -Δt, t₀ ] and [t₁ -Δt, t₁ ](Eq. 20 and 21) as follows: ##EQU14## Similarly, it can be proved (Eq.16-25) that:

    Δψ.sub.s1 =D(Δt.sub.1)Δψ.sub.s0  (29)

    i.sub.s,ave1 =D(Δt.sub.1)i.sub.s,ave0                (30)

    u.sub.s,ave1 =D(Δt.sub.1)u.sub.s,ave0                (31)

So the average values of the machine within the interval [t₁ -Δt, t₁ ]are obtained by delaying the average values of the interval [t₀ Δt, t₀ ]by Δt₁ (Eq. 17 and 28-31).

It can now be required that Eq. 12 is true for the time averages of thevalues corresponding to both the preceding and the later time interval.Introducing the values of the interval [t₀ -Δt, t₀ ] (Eq. 15, 20, 22 and24) and the values of the interval [t₁ -Δt, t₁ ] (Eq. 17 and 28-31) intoEq. 12 gives a pair of equations: ##EQU15## where ψ_(se) is an estimatefor the present-time stator flux (=ψ_(s) (t₀)), and

    a.sub.0 =Δψ.sub.s0 +(L.sub.s +R.sub.s τ.sub.r)i.sub.s,ave0 -τ.sub.r (u.sub.s,ave0 -σL.sub.s i'.sub.s,ave0) (33)

    b.sub.0 =Δψ.sub.s0 +σL.sub.s i.sub.s,ave0  (34)

    a.sub.1 =Δψ.sub.s +D(Δt.sub.1)a.sub.0      (35)

    b.sub.1 =Δψ.sub.s +D(Δt.sub.1)b.sub.0      (36)

The real and imaginary parts of the vectors ψ_(se), a₀, b₀, a₁ and b₁are indicated by the symbols ψ_(xe), ψ_(ye), a_(x0), a_(y0), b_(x0),b_(y0), a_(x1), a_(y1), b_(x1) and b_(y1), so that

    ψ.sub.se =ψ.sub.xe +jψ.sub.ye                  (37)

    a.sub.0 =a.sub.x0 +ja.sub.y0                               (38)

    b.sub.0 =b.sub.x0 +jb.sub.y0                               (39)

    a.sub.1 =a.sub.x1 +ja.sub.y1                               (40)

    b.sub.1 =b.sub.x1 +jb.sub.y1                               (41)

The equation pair 32 may now be written as follows:

    ψ.sub.xe.sup.2 +ψ.sub.ye.sup.2 -(a.sub.x0 +b.sub.x0)ψ.sub.xe -(a.sub.y0 +b.sub.y0)ψ.sub.ye +a.sub.x0 b.sub.x0 +a.sub.y0 b.sub.y0 =0(42)

    ψ.sub.xe.sup.2 +ψ.sub.ye.sup.2 -(a.sub.x1 +b.sub.x1)ψ.sub.xe -(a.sub.y1 +b.sub.y1)ψ.sub.ye +a.sub.x1 b.sub.x1 +a.sub.y1 b.sub.y1 =0(43)

In these equations the real and imaginary parts of the stator flux arethe only unknown parameters, and so they can be solved. To find thesolution, the corresponding sides of Eq. 42 and 43 are subtracted fromeach other, which gives

    c.sub.x ψ.sub.xe +c.sub.y ψ.sub.ye =d,             (44)

where

    c.sub.x =a.sub.x0 +b.sub.x0 -a.sub.x1 -b.sub.x1,           (45)

    c.sub.y =a.sub.y0 +b.sub.y0 -a.sub.y1 -b.sub.y1,           (46)

    d=a.sub.x0 b.sub.x0 +a.sub.y0 b.sub.y0 -a.sub.x1 b.sub.x1 -a.sub.y1 b.sub.y1(47)

Eq. 44 is then solved with respect to either ψ_(xe) or ψ_(ye),whereafter either ψ_(xe) or ψ_(ye), respectively, is eliminated from Eq.42. To avoid the division by zero, the solving is carried out withrespect to ψ_(ye) if |c_(y) |>|c_(x) |, otherwise with respect toψ_(xe).

The case where |c_(y) |>|c_(x) | will now be discussed. Thereby ψ_(ye)solved from Eq. 44 is introduced into Eq. 42, which gives

    q.sub.2 ψ.sub.xe.sup.2 +q.sub.1 ψ.sub.xe +q.sub.0 =0,(48)

wherein

    q.sub.0 =(a.sub.x0 b.sub.x0 +a.sub.y0 b.sub.y0)c.sub.y.sup.2 -(a.sub.y0 +b.sub.y0)c.sub.y d+d.sup.2,                              (49)

    q.sub.1 =(a.sub.y0 +b.sub.y0)c.sub.x c.sub.y -(a.sub.x0 +b.sub.x0)c.sub.y.sup.2 -2c.sub.x d,                      (50)

    q.sub.2 =c.sub.x.sup.2 +c.sub.y.sup.2                      (51)

Two solutions are now obtained from Eq. 44 and 48 for the flux:##EQU16##

Only one of these two solutions is "the right one", i.e. tends toestimate the actual stator flux. So the remaining problem is how toselect the right solution. In practice, the solutions according to Eq.53 and 54 are very remote from each other on the average, and so thesolution closer to the previous estimate is interpreted as the rightsolution (this will be discussed in greater detail below).

In the case |C_(x) |≧|C_(y) |, the solving of the flux estimate takesplace in a fully corresponding manner. Eq. 44 is thereby first solvedwith respect to ψ_(xe), whereafter ψ_(xe) is eliminated from Eq. 42. Inthis case, the solution is found by applying Eq. 48-54 so that thesubindices x and y are interchanged in the equations.

The above-described stator flux identification method is illustrated inFIG. 3, which is a block diagram illustrating the calculation of thecoefficient vectors a₀, b₀, a₁, b₁ presented in Eq. 33-36 and theselection of the right solution from the solutions of the equation pair32. The input parameters include the measured voltage u_(s) and currenti_(s), the machine parameters R_(s), L_(s), σL_(s), τ_(r) and theprevious flux estimate ψ_(se), prev. The output parameter will thus bethe stator flux estimate ψ_(se),id.

To calculate the coefficient a₀ in accordance with Eq. 33, the firstterm of the equation, i.e. the flux deviation Δψ_(s0), is calculatedfirst by first multiplying i_(s) and the stator resistance R_(s) witheach other in block 1 and then subtracting the obtained product from thevoltage vector u_(s) in block 2, the obtained different u_(s) -R_(s)i_(s) being then integrated in block 3 over the time interval [t₀ -Δt,t₀ ] by using t-(t₀ -Δt) as a weighting coefficient, whereafter theobtained integral is divided by Δt. To obtain the second term occurringin Eq. 33 , the stator resistance R_(s) and the rotor time constantτ_(r) are first multiplied with each other in block 4, and the resultingproduct is summed with the stator inductance L_(s) in block 5, and thesum so obtained is multiplied in block 7 by the time average i_(s),ave0of the current, which is calculated in block 6 by integrating i_(s) overthe interval [t₀ -Δt, t₀ ] and by dividing this integral by the durationof the interval, i.e. Δt. To obtain the third term included in Eq. 33,one first calculates the time average u_(s),ave0 of the voltage in block8 by integrating the stator voltage u_(s) over the time interval [t₀-Δt, t₀ ], and by dividing this integral by the duration of theinterval, i.e. Δt, and then subtracts in block 10 the product of theshort-circuit inductance σL_(s) and i'_(s),ave0 calculated in block 9from the obtained result, and finally multiplies the obtained differenceby the rotor time constant τ_(r) in block 11. The average derivative ofthe current, i.e. i'_(s),ave0, is obtained when a current signal delayedin block 13 by Δt is subtracted in block 12 from the present-timecurrent i_(s) and the obtained difference is divided in block 14 by Δt.The factors of the coefficient vector a₀ are then combined in block 15.

The coefficient vector b₀, in turn, is formed by first multiplying, inblock 16, the short-circuit inductance σL_(s) and the output of block 6with each other and then summing the output of block 3 to this productin block 17. The coefficient vectors a₁ and b₁ are thus formed on thebasis of the coefficient vectors a₀ and b₀ by delaying them in blocks 18and 19, respectively, by the operator D(Δt₁), and by summing the outputof block 22 to the delayed values in blocks 20 and 21. In block 22, theflux deviation Δψ_(s) is calculated by integrating the voltage u_(s)-R_(s) i_(s) generating the flux, that is, the output of block 2, overthe interval [t₀ -Δt₁, t₀ ].

In block 23, the equation pair 32 is solved on the basis of Eq. 32-54.Two solutions ψ_(se1) and ψ_(se2) are thus obtained, out of which thesolution closer to the value ψ_(se),prev of the previous calculation isselected as the final output parameter ψ_(se),id in block 24. Thiscomparison takes place in blocks 25-29 by forming the differencesbetween said solutions obtained in blocks 25 and 26 and the givenprevious stator flux estimate ψ_(se),prev, thus obtaining the vectors Δ₁and Δ₂. A parameter Δ₂₁ is calculated in block 29 as the differencebetween the absolute values determined in blocks 27 and 28. If Δ₂₁ ≧0,ψ_(se1) is selected as the flux estimate (indicated with the referenceψ_(se),id), otherwise ψ_(se2) is selected. Accordingly, the solution ofthe equation pair 32 closer to the previous flux estimate is interpretedas the right flux estimate.

In practice, the blocks of the figures in which definite integrals arecalculated may be realized by using e.g. FIR filters based on thesampling technique. Correspondingly, the delay blocks may be effectede.g. by shift registers.

In practice, the instantaneous accuracy of the estimate calculated bythe method of FIG. 3 is not very high; on the other hand, the accuracyof the estimate does not deteriorate as a function of time. In otherwords, the time average of the error of the estimate is constant andclose to zero, if the errors in the parameters needed in the method aresmall.

When the stator flux is calculated by Eq. 3, which is illustrated by theblock diagram of FIG. 4, the input parameters of the calculation includeonly the stator voltage u_(s), the stator current i_(s) and the statorresistance R_(s). In the block 30 of FIG. 4, i_(s) and R_(s) aremultiplied with each other, and the obtained product is subtracted inblock 31 from the stator voltage u_(s) to obtain the voltage generatingthe flux. To obtain the stator flux estimate ψ_(se), the output of block31 is integrated in block 32 with respect to time. In the procedure ofFIG. 4, the problems associated with the accuracy of the estimate arethe complete reverse of those occurring in connection with the procedureof FIG. 3. The flux estimate thereby follows rather accurately theactual stator flux over a short period of time, whereas a steady-stateerror accumulates in it in the long run. The steady-state errorincreases very forcefully with a decrease in the frequency and anincrease in the error present in the R_(s) estimate and in the measuringerrors.

Therefore, in the method according to the present invention, the methodsof FIGS. 3 and 4 are combined as illustrated in FIG. 5 so that thestator flux estimate produced by the method is as accurate as possibleboth over a short period of time and in the long run.

The input parameters of the method of FIG. 5 include the measured statorcurrent i_(s) and stator voltage u_(s) of the asynchronous machine. Inaddition, it is assumed that the stator inductance L_(s), theshort-circuit inductance σL_(s) and the rotor time constant τ_(r) areknown. The output parameter of the method is the stator flux estimateψ_(se) of the machine.

In FIG. 5, two estimates are calculated for the stator flux, one(ψ_(se)) of which is obtained by the integration method in accordancewith FIG. 4 and the other (ψ_(se),id) by the method of FIG. 3. Thecalculation of both estimates requires the stator current and statorvoltage and the stator resistance estimate (R_(se)), which is identifiedseparately in the respective method. The calculation of ψ_(se),id alsorequires L_(s), σL_(s) and τ_(r) and the previous flux estimate, whichis represented by the estimate ψ_(se) obtained by integration.

The idea is that the estimate calculated in blocks 30-32, correspondingto blocks 30-32 in FIG. 4, is integrated so as to correct it in thedirection of the estimate according to FIG. 3, the calculation of whichestimate is presented as a single block 33 in FIG. 5. A correction term(indicated with the reference Δψ_(se)) is first formed in block 34

    Δψ.sub.se =ψ.sub.se,id -ψ.sub.se,        (55)

and then one attempts to change ψ_(se) slowly in the direction of theterm. For this end, Δψ_(se) is first weighted by a constant coefficientw in block 35, and then summed in block 36 with a voltage u_(s) -R_(s)i_(s) to be integrated.

The identification of R_(s) utilizes the observation that when R_(se)comprises an error of a certain direction, this error affects the fluxcorrection according to Eq. 55 so that the scalar product of Δψ_(se) andi_(s) (indicated with the reference Δψ_(sei))

    Δψ.sub.sei =Δψ.sub.se ·i.sub.s =Re{Δψ.sub.se }Re{i.sub.s }+Im{Δψ.sub.se }Im{i.sub.s },(56)

which is calculated in block 37, is greater than zero if R_(se) isgreater than the actual value, and vice versa.

Therefore the described method aims at decreasing R_(se) if Δψ_(sei) >0and increasing it if Δψ_(sei) <0. This effect is produced when R_(se) iscalculated in block 39 by integrating Δψ_(sei) weighted by a negativeconstant coefficient (-w_(R)) in block 38, i.e.

    R.sub.se =∫(-w.sub.R Δψ.sub.sei)dt          (57)

The coefficient w_(R) is a positive constant which determines howrapidly R_(se) follows the variation in the actual stator resistance,which variation is caused e.g. by the temperature variations of thestator of the machine due to varying load. The smaller w_(R) is, themore slowly R_(se) may change. On the other hand, a high value of w_(R)causes great instantaneous variation in R_(se), which may result ininstability in the identification. In practice, w_(R) can be selected sothat it is extremely small, as R_(s) is actually able to change onlyvery slowly.

On selecting the coefficient w, it should be taken into account that thesmaller it is, the closer the estimate calculated by the method of FIG.5 is to the estimate calculated by the method of FIG. 4, that is, theestimate of FIG. 5 follows rather accurately the actual stator flux overa short period of time, whereas the steady-state error accumulating init over a longer period of time is the greater the smaller w is.Correspondingly, a high value of w causes the estimate to behavesimilarly as ψ_(se),id, that is, the steady-state error is small whilethe instantaneous error varies relatively greatly. In the selection ofw, one thus has to compromise so that both the instantaneous error andthe steady-state error are within acceptable limits.

I claim:
 1. A method for determining the stator flux of an asynchronousmachine when a stator current (i_(s)) and stator voltage (u_(s)) of theasynchronous machine are measured and a short-circuit inductance(σL_(s)), stator inductance (L_(s)) and rotor time constant (τ_(r)) ofthe machine are assumed to be known, wherein the product of the statorcurrent (i_(s)) and a stator resistance estimate (R_(se)) is determined,the obtained product is subtracted from the stator voltage (u_(s)), andthe obtained voltage difference is integrated with respect to time toobtain a first stator flux estimate (ψ_(se)), characterized in that themethod further comprises the steps ofdetermining a second stator fluxestimate (ψ_(se),id) on the basis of the stator voltage (u_(s)), thestator current (i_(s)), the short-circuit inductance (σL_(s)), thestator inductance (L_(s)), the rotor time constant (τ_(r)) and thestator resistance estimate (R_(se)); determining a stator fluxcorrection term (Δψ_(se)) as a difference between the first stator fluxestimate (ψ_(se)) and the second stator flux estimate (ψ_(se),id), andsumming the stator flux correction term (Δψ_(se)) weighted by a constantcoefficient (w) with said voltage difference.
 2. A method according toclaim 1, characterized in that the second stator flux estimate(ψ_(se),id) is determined by observing a differential equation (Eq. 12)defined on the basis of said machine parameters so as to describe theoperation of the machine in a neighbourhood (Δt) of two points of time(t₀, t₁) for obtaining an equation pair (Eq. pair 32) enabling themathematic solution of the stator flux estimate and by selecting thesecond stator flux estimate (ψ_(se),id) from the two solutions (ψ_(se1),ψ_(se2)) of the equation pair so that the selected flux estimate iscloser to the first stator flux estimate (ψ_(se)).
 3. A method accordingto claim 2, characterized in that the observation in the neighbourhoodof the second point of time (t₁) is carried out by delaying the valuesobtained in the neighbourhood of the first point of time (t₀) by theamount of time (Δt₁) between the points of time and by summing theobtained values with a flux change (Δψ_(s)) over said time interval(Δt₁), the flux change being determined by integrating said voltagedifference over said time interval.
 4. A method according to claim 1,characterized in that the stator resistance estimate (R_(se)) isdetermined on the basis of the stator current (i_(s)) and said statorflux correction term (Δψ_(se)).
 5. A method according to claim 4,characterized in that the determination of the stator resistanceestimate (R_(se)) comprises the steps ofdetermining the scalar product(Δψ_(sei)) of the stator flux correction term (Δψ_(se)) and the statorcurrent (i_(s)); weighting said scalar product (Δψ_(sei)) by multiplyingit by a negative constant coefficient (-w_(R)); and integrating saidweighted scalar product with respect to time.